When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Euler's Letter to Cramer

The full text of Euler's letter to Cramer of October 20, 1774 is reproduced here in English translation.

Sir,

As I have not yet seen my work, which has just come off the press,¹ I am infinitely obliged to you for the particular trouble you have taken with the corrections. But great though my obligation to you may be, so much greater should be my sympathy for the precious time you have expended, and the scientific community [les Scavans] should be very displeased with me for causing you to have turned away from your usual occupations, so highly esteemed by all. It is because of this consideration that I completely approve of the reply you have made to Mr. Bousquet, in refusing your assistance with respect to proofreading my work,² which he wishes to publish, not doubting for a moment that he would never find a man as qualified for the task as you in Lausanne. I have learned with great pleasure that you have composed a piece on the same material³ and, as I am extremely curious to see it, I add my wishes to those of Mr. Bousquet to encourage you to publish it. In my opinion, these matters have, by and large, not yet been properly explained and I do not doubt that you have clarified a great number of situations that have eluded me, as well as others who have written on the subject. One finds there questions so thorny, that one must apply to them all possible attention so as not to fall into error, as happened to me in my explanation of the cuspidal point of the second kind.⁴ Mr. the Marquis de l'Hopital showed that there are in fact curves endowed with such a point,⁵ but Mr. Gua de Malves holds that the two branches of a curve which form the point always extend to the other side so that, according to him, this point is nothing but the intersection of two branches, which cross in an infinitely small angle.⁶ These arguments convinced me that he was right, as you no doubt noted in looking over my work. But since then, I have recognized quite clearly that I was mistaken on this and that there actually are curves that have such a cusp point by itself, one that cannot be regarded as the infinitely close intersection of two branches. Even in the fourth order there is a curved line of this kind, whose equation is⁷ \[y^4 - 2xy^2 + xx = x^3 + 4yxx,\] which simplifies to \[y=\sqrt{x} \pm \sqrt[4]{x^3}.\]

The reason I was mistaken is that I believed that this curve ought to have a diameter,⁸ since \(\sqrt{x}\) may take a negative value as well as positive, but since the other term \(\sqrt[4]{x^3}\) is equal to the first one, \(\sqrt{x},\) multiplied by its own square root \(\sqrt{\sqrt{x}},\) one sees clearly we may not take the first \(\sqrt{x}\) to be negative, without the other \(\sqrt[4]{x^3}\) becoming imaginary.  And in fact, if we give the \(\sqrt{x}\) term the \(-\) sign, then the equation

\[y=-\sqrt{x} \pm \sqrt[4]{x^3}\]

is resolved as

\[y^4 - 2xy^2 + xx = x^3 - 4yxx,\]

which is not the same curve in the same position. As I do not have a copy of my manuscript here, I beg you to add a little note at this location, if you have not already returned home.

I have seen that Mr. Maclaurin already had the same doubt concerning the number of points which determine curves of a given order:  he says that to determine a line of the third order, the number of nine points may be too small, yet still the number of ten is too great, which in my opinion is an overt contradiction. The afore-mentioned Braikenridge is also absolutely mistaken in holding that a line of order \(n\) may be described by \(n^2+1\) points and it is a disputed truth, as you have very well remarked, that this number is but \(\frac{nn+3n}{2}.\)  Furthermore, one may not doubt that two curved lines, one of which is of order \(m\) and the other of order \(n,\) may intersect in \(mn\) points, though you will be the first to have given a perfect proof of this truth, for I freely admit that my proof is all but complete. At first, all of these reflections only served to bring to my attention the difficulties of the case, which you were so good as to propose to me. However, I finally found the solution to this doubt, with which I hope you will be satisfied. I say, then, that although it is indeed true that a line of order \(n\) be determined by \(\frac{nn+3n}{2}\) points, this rule is nevertheless subject to certain exceptions.  For although the general equation of lines of order \(n\) has \(\frac{nn+3n}{2}\) coefficients to be determined, it may happen that such a number of equations, which we draw from the same number of given points, is not sufficient for this effect: this is evident, when two or several of these equations become identical. In such a case, one finds after having reduced the matter to the determination of the final coefficient, the value of this is expressed by a fraction, whose numerator and denominator both become \(=0.\) I conceive therefore, that this inconvenience will take place when the nine points, which ought to determine a line of the 3rd order, are disposed such that two curved lines of this order may be drawn through them. In this case, the nine given points, since they contain two identical equations, are worth but 8, and we may then add the tenth point in order to render the problem determined. We may clarify this article to our further satisfaction by considering lines of the second order, for the determination of which 5 points may not always be sufficient. For when all the five points are arranged on a straight line so that they give, for example, these equations⁹

all of the coefficients of the general equation \(\alpha yy + \beta xy + \gamma xx + \delta y + \epsilon x + \zeta = 0\) will not be determined, for after having introduced all of the given determinations, we are brought to this equation \(\alpha yy - (\alpha + \gamma)xy + \gamma xx  + \delta y - \delta x = 0,\) so that there still remain two coefficients to be determined. If from the five given points there had been but 4 arranged in a straight line, then there would remain but one coefficient to be determined. From this, one easily understands that if the nine points, from which one ought to draw a line of the third order, are at the same time the intersections of two curved lines of this order, then, after having completed all of the calculations, there must remain in the general equation for this order an undetermined coefficient, and beginning from this case not only two, but an infinity of lines of the 3rd order may be drawn from the same nine points.

The use, which you have made of continued fractions in dioptrics is admirably beautiful and I am extremely obliged to you for the theorem, which you have communicated to me. I am charmed that you recognize, along with me, that this material is of great use in mathematics and that it is quite worthy of attention. It is not only arithmetic that can draw much profit from it, but also the integral calculus, as I made known in several pieces on this subject that I left in Petersburg, one of which¹⁰ has already been published in the ninth volume of the Comm.

It is already a long time since Mr. Bousquet wrote to tell me that you had the kindness to send me a copy of the Works of Mr. James Bernoulli, which was shipped here along with a quantity of books for Mr. Neaulme. But since this latter was not willing to accept the package, I have received nothing. Had it not been for this, I would not have failed to thank you infinitely. I am therefore embarrassed that I do not find myself in a position to show my gratitude except in words, but rest assured, that should an opportunity present itself for me to render you service, I will employ all of my energy to discharge my obligation. I have the honor of being, with the most perfect esteem,

      Sir,

      Your very humble and very obedient servant L. Euler

      Berlin this 20 October 1744

Notes on the Translation of Euler's Letter to Cramer

¹ Here Euler referred to Methodus inveniendi lineas curvas [Euler 1744a], published in Lausanne by Bousquet in 1744.
² Here Euler referred to Introductio in analysin infinitorum [Euler 1748], published in Lausanne by Bousquet in 1748.
³ Cramer's Introduction a l'analyse des lignes courbes algébraique [Cramer 1750], published by Bousquet in 1750, and volume 2 of Euler's Introductio [Euler 1748] both dealt with the theory of equations.
⁴ For more on the cuspidal point of the second kind and the importance of the 4th degree equation that follows, see [Bradley 2006].
⁵ See [L'Hôpital 1696].
⁶ See [Gua de Malves 1740].
⁷ The four equations in this paragraph were actually written by Euler as in-line equations. We have set them as displayed equations for greater clarity.
⁸ That is, Euler thought the curve was symmetric about the \(x\)-axis.
⁹ In modern notation, Euler considered the points \((0,0), (1,1), (2,2), (3,3),\) and \((4,4).\)
¹⁰ Euler referred to E71, "De fractionibus continuis dissertatio," which was presented to the St. Petersburg Academy on March 7, 1737. However, volume 9 of the Commentarii academiae scientiarum imperialis Petropolitanae, for the year 1737, did not actually appear until 1744.